Rigidity theorem for degenerated singular points of germs of holomorphic vector fields in the complex plane (Q1868378)

The paper addresses the class \(V_n\) of germs of holomorphic vector fields in \((C^2,0)\) with vanishing \((n-1)\) jet at zero. We recall that two germs \(v,w\in V_n\) are called orbitally analytically equivalent, if one of them can be transformed into the other by a local holomorphic change of variables followed by the multiplication of an invertible holomorphic germ of a function (by the multiplication of a formal power series with a nonzero constant term). The major result here proves that the formal equivalence of two generic germs in \(V_n\) implies their analytic equivalence. The result is analogous to the result in the paper due to \textit{S. M. Voronin} [Yu. S. Il'yashenko (ed.) et al., Differential equations with real and complex time. Collection of articles. Moskva: Nauka. MAIK Nauka. Tr. Mat. Inst. Steklova, 213, 35--55 (1997; Zbl 0910.34040)]. The theorem is clearly stated as: Formal equivalence of generic germs of class \(V_n\) implies their analytic equivalence. Its analogously stated form of the theorem is that formal orbital equivalence of generic germs of the class \(V_n\) implies their analytic orbital equivalence. The authors present the proof in an extensive set of lemmas including some very good diagrams to illustrate the significant curves. The paper is very well written and clearly proves the result.